Chen, Y.-H., Looi, C.-K., Lin, C.-P., Shao, Y.-J., & Chan, T.-W. (2012). Utilizing a Collaborative Cross Number Puzzle Game to Develop the Computing Ability of Addition and Subtraction. Educational Technology & Society, 15 (1), 354–366.
Utilizing a Collaborative Cross Number Puzzle Game to Develop the Computing Ability of Addition and Subtraction 1
Yen-Hua Chen1, Chee-Kit Looi2, Chiu-Pin Lin3, Yin-Juan Shao2 and Tak-Wai Chan4
Department of Computer Science and information Engineering, National Central University, Taiwan // 2Learning Sciences Lab, National Institute of Education, 1 Nanyang Walk, Singapore 637616 // 3Graduate Institute of eLearning Technology, National Hsinchu University of Education, Taiwan // 4Graduate Institute of Network Learning Technology, National Central University, Taiwan //
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ABSTRACT While addition and subtraction is a key mathematical skill for young children, a typical activity for them in classrooms involves doing repetitive arithmetic calculation exercises. In this study, we explore a collaborative way for students to learn these skills in a technology-enabled way with wireless computers. Two classes, comprising a total of 52 students in Grade 4 (ages 10 or 11) participated in the study. They used the Group Scribbles software to run an adapted version of the “Cross Number Puzzle” that was designed with the “feedback” mechanism to assist students’ problem solving. In one class, students played the game individually and in the other class, students played the game collaboratively. The low-ability students in the collaborative class were found to have made the most significant progress in arithmetic skills through playing this game. Three dominant interactive collaboration patterns, one contributing to productive interactions and two to less productive interactions, were also identified in the students’ collaboration.
Keywords Computer-supported collaborative learning, feedback mechanism, Cross Number Puzzle, computing ability
Introduction Computational skills in addition and subtraction are an important component in any mathematics curriculum for young children. In seeking to design more interesting learning experiences for children to learn these skills, we designed and implemented a learning activity which is an adapted version of the “Cross Number Puzzle” on a technology platform. The aim of this game is to promote the flexible application of addition and subtraction skills, and to enhance children’s capacity to build up their arithmetic skills progressively. We conducted a study on two classes of students to explore different collaborative learning patterns that involved students working together on solving arithmetic problems. We also examined differences between individual learning and collaborative learning. A variety of educators have classified operating addition and subtraction story problems into four problem types: change, combine, compare and equalizer (Carpenter, Hiebrt, and Moser, 1981; Fuson, 1992; Gustein and Romberg, 1995). English (1998) points out that change and combine are easier while take-away and compare are more difficult challenges for elementary school students. In an arithmetic equation, any of the three numbers could be the unknown number. We adopted this widely used method in our study. Fuson (1992) defines these three types of “change” (placeholder) as: Missing End, Missing Change, and Missing Start. Van de Walle (2001) also classifies the type of “change” into three types: result-unknown, change- unknown and initial-unknown. These three types of problems present different levels of difficulty to the students. In Tompson’s study (1983), the initial unknown is most difficult. If the student applies the direct modeling strategy by using counters or tally marks to model directly the action or relationships described in the problem (Carpenter et al., 1993), he or she always does not know how many counters to be put down to begin with. Table 1 illustrates the three levels of change types in story problems. Level I is when the result number is unknown, Level II is when the change number is unknown, and Level III is when the initial number is unknown (Carey 1991; Peterson et al, 1991). With these three levels in Table 1 in mind, we designed our system to incorporate five stages of problem posing to the students (Table 2): In stage 1, the student is required to derive the answer of an arithmetic expression (result number unknown), inculcating the skills of basic addition and subtraction, for example: 3 ± 2 = □. In stage 2, the arithmetic operator is removed. Students were required to understand the concept of arithmetic operator, for example: 3 □ 2 = 5. In stage 3, the change amount is removed, for example: 3 ± □ = 5. ISSN 1436-4522 (online) and 1176-3647 (print). © International Forum of Educational Technology & Society (IFETS). The authors and the forum jointly retain the copyright of the articles. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear the full citation on the first page. Copyrights for components of this work owned by others than IFETS must be honoured. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from the editors at
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In stage 4, the initial amount is removed, for example: □ ± 3 = 5. In stage 5, both the initial and the change numbers are removed. It is more difficult as the sentence includes two variables: change amount and initial amount unknown. For example: □ ± □ = 5. Change Types Result number unknown Change number unknown
Initial number unknown
Level of difficulty Level 1 Level 2 Level 3 Level 4 Level 5
Table 1. Three levels of “Change” types in story problems Join(Add to) Separate(Take from) John has 5 cookies; Marry gave 5 cookies. John has 8 cookies; he gave Marry 5 How many does John have now? cookies. How many does John have now? Standard sentence: A + B = □ Standard sentence: A – B = □ John has 5 cookies; Marry gave some John has 8 cookies; he gave Marry some cookies, and now John has 8. How many cookies, and now John has 5. How many did did John get from Marry? John give to Marry? Standard sentence: A - □ = B Standard sentence: A + □ = B John has some cookies; Marry gave 5 John has some cookies; he gave 5 cookies to cookies, and now John has 8. How many Marry, and now John has 3. How many cookies did John has before Marry gave cookies did John has before he gave some to him some? Marry? Standard sentence: □ + A = B Standard sentence: □ – A = B Table 2. Level of difficulty in the design Description Result number unknown － basic skill practice Remove operator－between basic skill practice and comprehension application Change number unknown add-to or subtraction－comprehension application Initial number unknown add-to and subtraction－comprehension application Change number unknown and Initial number unknown, addend or summand type －the most difficult level
Example A ± B =□ A□B=C A± □=B □ ±A= B □ ±□ =A
Design of the “Feedback” system A ‘feedback’ mechanism was introduced to the game design in this study. Feedback is considered to have strong impact on the learning process and result (Kulhavy and Stock, 1989; Bangert-Drowns, Kulick, Kulik and Morgan, 1991; Balacheff and Kaput 1996). Appropriate feedback can lead learners to focus on key elements of learning. They can always adjust their learning strategies to try to close the gap between their actual performance and the learning goals. They reflect on their learning by a self-monitoring feedback loop. Hence, they can change their learning strategies in the follow-up learning and seek a better way of learning (Alexander and Shin, 2000). The learner understands the gap or difference between her own concept and the target concept, and the perceived difference can provide information for her to amend her ideas and knowledge, thus enacting a process of signal feedback. Schmidt (1991) proposes that feedback is the result of a series of actions. It represents the personal response or reaction to the information they received. The feedback itself is a problem solving process that checks the performance of action to improve a person or a group. In technology-enabled learning, feedback is typically provided as messages shown to students after their responses (Cohen, 1985). Siedentop (1991) points out that feedback can promote the interaction between the teacher and the learners. Teachers can provide feedback to students in terms of their actions and performance, which enable them to know or to amend their understandings and may boost their enthusiasm for learning (Keh, 1992). There are three forms of feedback: immediate feedback, summary feedback and compromise feedback (Schmidt and Wrisberg, 2000). Collins, Carnine, and Gersten (1987) point out three levels of feedback messages: (1) little feedback: just show the answer is right or wrong; (2) basic feedback: if answer is not correct then show right answer; and (3) descriptive feedback: give some hints to learner to help her to obtain the right answer. Descriptive feedback can increase the motivation for learners to try or solve new tasks and new problems. The feedback mechanism provided by software systems mainly involves five levels such as: no feedback, knowledge of response, knowledge 355
of correct response, answer until correct, and elaboration feedback, as summarized by Sales (1998). In our research, the feedback mechanism is designed as follows (Figure 1): Flow chart Knowledge of response
Answer until correct
Knowledge of correct response
Description Feedback to the student regardless of whether her response is correct or incorrect Set the upper limit of number of incorrect attempts before providing a new clue
Point out what should be amended
Answer until correct, AUC
Set the upper limit of number of incorrect attempts before providing a give new clue
Elaboration feedback, EF
Provide the detailed answer, explain and elaborate the answer
Figure 1. Five-level feedback mechanism Progressive feedback is provided in a step by step manner in this five-level mechanism. In the ‘knowledge of response’, the student get a response whatever she replies correctly. Then a constraint on the upper limit of incorrect attempts is invoked to allow some trial and error. If the student fails after a number of trials, a clue or a suggestion would be offered to her for another round of trial and error. In the ‘elaboration feedback’, the correct answer together with detailed explanations will be provided.
Methodology Fifty two students in Grade 4 (ages 10 or 11) participated in our study. They learned some basic addition and subtraction since Grade 1 but they needed to connect this to the new concepts and skills required for Grade 4 mathematics. In this research we explored the effects of “Cross number puzzle” game applied in learning, which was designed to provide the feedback mechanism. We had two experimental classes: students in Class A played the “Cross number puzzle” game in small groups, and students in Class B played the game individually. All students were grouped according to their average scores of the previous three tests in this term. Using percentile ranking, those students with the percentile rank of score over 73% were classified as high-math achievers; those with percentile rank of score between 27% and 72% were classified as medium-math achievers, and those with percentile below 26% were classified as low math achievement. Students in class A were divided into homogeneous groups with three per group. 6 students in the high-achiever group form two groups. Three medium-achiever groups comprised 9 students and another three low-achiever groups comprised 9 students. We utilized Group Scribbles (GS) as the platform for the game, and conducted analysis of the collaborative work within these groups. GS is a computer-supported collaborative learning system developed by SRI International to conduct small-group collaborative concept mapping activities (Chaudhury, et al, 2006; Looi, Chen and Ng, 2010). Each student has a Tablet PC which has a screen divided into upper and lower frames (Figure 2). The lower frame is for individual cognition, that is, the student sketches or types her answer individually. The upper frame is a shared space (public board) in which the students show all of their individual answers, and work together as a group. They can even check the work from other groups by clicking the button on the top right corner (see Figure 2). The teacher can monitor their process of learning and provide appropriate guidance.
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Public Board
Change to another group’s results
Private Board
Figure 2. Spaces for individual and public cognition work Figure 2 shows the interface of “Cross Number Puzzle” in GS. We designed questions ranging from the easy to the difficult in terms of the five levels of difficulty. When the students complete the calculation, they can fill in the answer box and press OK button under the question area to submit. If the answer is correct, there will be a brief description of the key points. If the answer is wrong, the system will execute a step-by-step tip based on the number of errors from the user inputs (Figure 8, a popup box with tips shown in individual area). The action repeats until the maximum number of errors reaches the upper limit. Then the system will show the correct answer and the methods of problem-solving. Four different types of questions were shown below (Figure 3 to Figure 6) in the “Cross Number Puzzle”.
Figure 3. Question type 1
Figure 4. Question type 2
Figure 5. Question type 3
Figure 6. Question type 4
Figure 7. Interface of individual and public work
Figure 8. Tips
Figure 9. Calculation process 357
Figure 7 shows a screenshot of one student’s calculation process. Questions appeared in the private board and the individuals could paste their sketches to the public board. Figure 9 provides the screenshot of the calculation process in a four-member group, in which each individual pasted his/her sketches using a different color. The study for both classes lasted for four weeks. In the first week, a session for 30-minute pre-test and 20-minute training was executed. Students were asked to be familiarized with GS and the operation of the game with simple exercises. In the second week, the game was played in one lesson lasting for 60 minutes, followed by a 30- minute post-test and a 20-minute questionnaire in the third week. In the fourth week we interviewed the teachers and students. A pre-activity and three learning activities were included in this study. In the game playing session, three tasks were designed and implemented in collaborative group (Class A) and individual group (Class B) separately. Students were asked to fill in the operator in an arithmetic equation in Activity 1. Activity 2 is about filling in the unknown number while in Activity 3, students were asked to estimate using trialand-error methods to solve the problem. The only difference between these two classes is that students in Class B played the game individually but students in Class A played it collaboratively. Figure 10 below gives an example of a game screenshot of a collaborative group with four members. Group members could use their private boards for sketches and confirmatory calculations. Then they could post their sketches or results to the public board. Figure 11 shows an example of a student’s work from a student who worked individually. We can inspect this student’s solution path to identify and understand his working strategy.
Figure 10. Change number unknown and initial number unknown exercise – a group’s work
Figure 11. Change number unknown and initial unknown exercise – a student who worked individually
Findings We analyzed students’ scores in pre and post-tests, collected questionnaires, did video-recording of their activities in classes and tracked their screens in the process of game playing. The pre-test and post-test have the same questions but they are ordered differently. We also verify the item discrimination index and had the questions validated by a experienced mathematics teacher. Results of the assessment of computing ability We administered pre-tests and post-tests and performed independent sample t-test of two classes on their results. Table 3 shows the pre and post test results of Class A. Students in Class A have higher average scores in the posttest. Their average score was increased by 13.00, from 50.29 in pre-test to 63.29 in post-test (p=0.002
